The solution set of the inequation `|(1)/(x)-2| lt 4`, is

Options

A. `(-oo, -1//2)`
B. `(1//6, oo)`
C. `(-1//2, 1//6)`
D. `(-oo, -1//2) cup (1//6, oo)`

Explanation

To solve the inequation \(\left|\frac{1}{x} - 2\right| < 4\), we can break it down into two separate inequalities. Here’s the step-by-step solution:

### Step 1: Rewrite the Absolute Value Inequality
The inequality \(\left|\frac{1}{x} - 2\right| < 4\) can be expressed as:
\[ -4 < \frac{1}{x} - 2 < 4 \]

### Step 2: Split into Two Inequalities
This gives us two inequalities to solve:
1. \(\frac{1}{x} - 2 < 4\)
2. \(\frac{1}{x} - 2 > -4\)

### Step 3: Solve the First Inequality
For the first inequality:
\[ \frac{1}{x} - 2 < 4 \]
Adding 2 to both sides:
\[ \frac{1}{x} < 6 \]
Taking the reciprocal (and reversing the inequality since \(x\) must be positive):
\[ x > \frac{1}{6} \]

### Step 4: Solve the Second Inequality
For the second inequality:
\[ \frac{1}{x} - 2 > -4 \]
Adding 2 to both sides:
\[ \frac{1}{x} > -2 \]
Taking the reciprocal (and reversing the inequality since \(x\) must be positive):
\[ x < -\frac{1}{2} \]

### Step 5: Combine the Results
From the two inequalities, we have:
1. \(x > \frac{1}{6}\)
2. \(x < -\frac{1}{2}\)

### Step 6: Exclude the Point Where \(x = 0\)
Since \(x\) cannot be zero (as it would make \(\frac{1}{x}\) undefined), we conclude that the solution set is:
\[ x \in (-\infty, -\frac{1}{2}) \cup (\frac{1}{6}, \infty) \]

### Final Solution
The solution set of the inequation \(\left|\frac{1}{x} - 2\right| < 4\) is:
\[ (-\infty, -\frac{1}{2}) \cup (\frac{1}{6}, \infty) \]